17_Design of Mufflers and Silencers2

8/21/2019 17_Design of Mufflers and Silencers2

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Design of Mufflers and Silencers

Slides to accompany lectures in ME 599/699:

Vibro-Acoustic Design in Mechanical Systems 2003 by A. F. Seybert

Department of Mechanical Engineering

University of Kentucky

Lexington, KY 40506-0108

Tel: 859-257-6336 x 80645Fax: 859-257-3304

[email protected]


2/40

Dept. of Mech. EngineeringUniversity of Kentucky 2


ME 599/699 Vibro-Acoustic Design

Goals of the Lecture

To understand how the components and geometry of amuffler (silencer) determine its performance

To design a muffler for a specific attenuation andfrequency range

To learn and use muffler terminology


3/40




Types of Mufflers

1. Dissipative (absorptive) silencer:

Sound is attenuated dueto absorption (conversion

to heat)

Sound absorbing material(e.g., duct liner)

Duct or pipe


4/40




Types of Mufflers (cont.)

2. Reactive muffler:

Sound is attenuated by reflection andcancellation of sound waves

Compressor discharge details

40 mm


5/40




Types of Mufflers (cont.)

3. Combination reactive and dissipative muffler:

Sound is attenuated by

reflection and cancellation ofsound waves + absorption ofsound

Sound absorbing material

Perforated tubes


6/40




Performance Measures Transmission Loss

Transmission loss (TL) of the muffler:

Wi

Wr

Wt AnechoicTerminationMuffler

TL dB Log WW

i

t

( )=10 10


7/40




Performance Measures Insertion Loss

IL (dB) = SPL1 SPL2

Insertion loss depends on : TL of muffler Lengths of pipes Termination (baffled vs. unbaffled) Source impedance

Muffler

SPL1

SPL2


8/40




Example TL and IL

24

12

122 6Source

-50

-40

-30

-20

-10

0

10

20

0 200 400 600 800 1000

Frequency (Hz)

TL

and

IL

(dB)

Insertion Loss

Transmission Loss

Pipe resonances

Inlet Pipe Outlet Pipe

Expansion Chamber Muffler


9/40




Zeffort variable

flow variable

W = effort variable x flow variable *

SYSTEM

Flow variable

Effort variable

+

_

The Concept of Impedance

Generalized impedance of a systemZ and power supplied W :

PAV AS = P2 AS /ZP/VVelocity (V)Pressure (P)Acoustic

T/Q

P/Q

F/V

E/I

Impedance Z

PAQ = P2/ZVolume Flow Rate (Q)Pressure (P)Fluid

TAQ = T2/ZHeat Flow Rate/Deg(Q)

Temperature(T)

Thermal

FAV = F2/ZVelocity (V)Force (F)Mechanical

EAI = E2/ZCurrent (I)Voltage (E)Electrical

Power supplied WFlow VariableEffort VariableSystem Type

* For acoustic systems we must multiply by the area S


10/40




Network Interpretation: Electrical to Acoustical

ZV

P

source loadZI

E

source load

Electrical System Acoustical System


11/40




Source VAny acoustic

system

V

P(sound

pressurereaction)

Zt

z

P

V r jx= = + zP

V r xtt

t

t t= = +Input or loadimpedance

Terminationimpedance

Acoustic System Components


12/40




Summary 1

Dissipative mufflers attenuate sound by convertingsound energy to heat via viscosity and flow resistance this process is called sound absorption.

Common sound absorbing mechanisms used indissipative mufflers are porous or fibrous materials orperforated tubes.

Reactive mufflers attenuate sound by reflecting aportion of the incident sound waves back toward thesource. This process is frequency selective and mayresult in unwanted resonances.

Impedance concepts may be used to interpret reactivemuffler behavior.


13/40




The Helmholtz Resonator

Named for:

Hermann von Helmholtz, 1821-1894, Germanphysicist, physician, anatomist, and physiologist.

Major work: Book, On the Sensations of Tone asa Physiological Basis for the Theory of Music,1862.

von Helmholtz, 1848


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Helmholtz developed a set ofresonators for studying the auditory

response of humans to tones.

Modern applications:

fundamental resonance of stringed instruments base-reflex (ported) loudspeakers

muffler components

History


15/40




Lumped Parameter Model

Mx Kx PS x j v v

j

j M

K

v PS

z P

v j

S M

K K

M c

S

L V

B B

B

B B

B

B B

B

&& &&

'

+ = = =

=

= =

= =

x

when

1

0

Kc S

V

M S L

o B

o B

=

=

2 2

'

F = PSB

x

V

SB

L

L is the equivalentlength. Losses due toviscosity in the neck and

radiation are neglected.


16/40




Example HR Used as a Side Branch

V = 0.001 m3

L = 25 mmSB= 2 x 10

-4 m2

S= 8 x 10-4 m2

fn = 154 Hz

Anechoic termination

TL dBc S

L S c VB( ) log

/

'/ /= +

10 12

10 2

2

0

5

10

15

20

0 50 100 150 200 250 300

Frequency (Hz)

TL

(dB)

35 Hz


17/40




A Tuned Dynamic Absorber

K1

M1xF

K1

M1xF

K2

M2

" "Tune K

M

K

M2

2

1

1

=

Original System

Tuned Dynamic Absorber

T/T1

|x/F|

Original system

Tuned dynamic absorberM2/M1=0.5


18/40




Resonances in an Open Pipe

P = 1 Pa

Lp= 1 msource

First mode

Second Mode

11

12343

2 1 1715= = = =L

c

f fp ( )

. Hz

22

2

343

1

343= = = =Lc

f

fp Hz

etc.


19/40




SPL at Pipe Opening No Resonator


20/40




Example HR Used as a Side Branch

V = 750 cm3

L = 2.5 cm (L= 6.75 cm)DB= 5 cm (SB= 19.6 cm

2)D= 10 cm (S = 78.5 cm2)

fn = 340 Hz

Anechoic termination

TL dBc S

L S c VB( ) log

/

'/ /= +

10 12

10 2

2


21/40




SPL at Pipe Opening with Resonator


22/40




Network Interpretation

z z z

z zB A

B A

=+

Can we make ZB zero?

zAV

P

zB

z

z zA

zB(any system)

z Pv

jS

M K KM

c SL VB B B

B= =

= =

1 0

when'

(Produces a short circuit and P is theoretically zero.)


23/40




The Quarter-Wave (QW) Resonator

z j c L c L c n

n cL

f nc

L

nc

f n

B o

n

n

= = = =

=

= = =

cot( / ) / / , , ...0 2 13 5

2

4 4 4

when n

or L

The Quarter-Wave Resonator has an effect similar to the HelmholtzResonator:

( )( )

+=2

22

104

4tan10

b

b

SS

SSkLLogTL

zB

L

S

Sb


24/40




Summary 2

The side-branch resonator is analogous to thetuned dynamic absorber.

Resonators used as side branches attenuate

sound in the main duct or pipe.

The transmission loss is confined over arelatively narrow band of frequencies centeredat the natural frequency of the resonator.


25/40




The Simple Expansion Chamber

TL Log kL mm

kL= + +

10

1

4 4

110

2 2 2cos ( ) sin

18

2 26

where m is the expansion ratio (chamber area/pipearea) = 9 in this example and L is the length of the

chamber.

0

5

10

15

20

25

30

0 100 200 300 400 500 600 700 800

Frequency (Hz)

TL

(dB)


26/40




QW Tube + Simple Expansion Chamber

0

5

10

15

20

25

30

0 100 200 300 400 500 600 700 800

Frequency (Hz)

TL

(dB)

2

918

2 26


27/40

Dept. of Mech. EngineeringUniversity of Kentucky

27



Extended Inlet Muffler

18

2 269

0

5

10

15

20

25

30

0 100 200 300 400 500 600 700 800

Frequency (Hz)

TL

(dB) ( same for

extendedoutlet )


28/40


28



Two-Chamber Muffler

0

10

20

30

40

50

0 100 200 300 400 500 600 700 800

Frequency (Hz)

TL

(dB)

9 9

4 6


29/40


29



Source

Engine

PumpCompressor(intake or exhaust)

Area change

Expansion chamber

Helmholtz Resonator

Quarter-wave resonator

termination

Complex System Modeling

We would like to predict the sound pressure level at the termination.


30/40


30



The sound pressure p and the particle velocity v are the acoustic state variables

any acousticcomponent

2

1

p2, v2

p1, v1

For any passive, linear component:

p Ap Bv

v Cp Dv

or

p

v

A B

C D

p

v

2 1 1

2 1 1

2

2

1

1

= += +

=

Basic Idea

Transfer, transmission, or four-pole matrix

(A, B, C, and D depend on the component)


31/40


31



Combining Component Transfer Matrices

[ ] [ ] [ ][ ] [ ]

[ ]

[ ]

TA B

C D

pv

T T T T pv

T pv

TA B

C D

i

i i

i i

n

nn i system

system

system system

system systemi x

=

=

=

=

2x2

th = transfer matrix of i component

3 2

1

1

1

1

2 2


32/40


32



Performance Measures Transmission Loss

Transmission loss (TL) of the muffler:

TL dB Log W

W

i

t

( )=10 10

TL Log S

S A cC

B

c

S

S D

oo

=

+ + +

10 1

4102

1

1

2

2

Wi

Wr

Wt AnechoicTermination

2 1

A B

C D


33/40


33



The Straight Tube

p2, v2 p1, v1

S

LA

B

(x = 0) (x = L)

p x Ae Be v xjk c

dp

dx

p p A B

v vA B

c

p L p Ae Be

v L vAe Be

c

p p kL v j c kL

v p j c kL v kL

p

v

kL j c kL

j c kL kL

jkx jkx

o

o

jkL jkL

jkL jkL

o

o

o

o

o

( ) ( )

( )

( )

( )

( )

cos ( )sin

( / )sin cos

cos sin

( / )sin cos

= + =

= = +

= =

= = +

= =

= +

= +

=

1

0

0

2

2

1

1

2 1 1

2 1 1

2

2

p

v

1

1

Solve for A, Bin terms of p1, v1then put into

equations for p2,v2.

(note that the determinant A1D1-B1C1 = 1)

must have plane waves


34/40


34



Straight Tube with Absorptive Material

p

v

k L jz k L

j z k L k L

p

v

c

c

2

2

1

1

=

cos ' sin '

( / )sin ' cos '

L

k,zc

(complex wave number andcomplex characteristic

impedance)


35/40

Dept. of Mech. Engineering

University of Kentucky 35



Area Change

S2

S1

2 1

p p

S v S v

2 1

2 2 1 1

==

p

v S S

p

v

2

2 1 2

1

1

1 0

0

=

/


36/40





Expansion Chamber Muffler

L

S SSstraight

tube

area changes

[ ]'/

cos sin

( / )sin cos / '

[ ]cos ( / ) sin

( / )sin cos

TS S

kL j c kL

j c kL kL S S

TkL j c m kL

m j c kL kL

o

o

o

o

=

=

1 0

0

1 0

0

(m = S/S is called the expansion ratio of the muffler)


37/40





Expansion Chamber Muffler - Example

TL Log S

S A cC

B

c

S

S D

TL Log kLm

kL

oo

=

+ + +

= + +

10 1

4

10 1

4 4

1

10

2

1

1

2

2

10

2 2 2

cos (m ) sin

Recall:

18

2 26

(m = 9)


38/40





Transfer Matrix for a Side-Branch

SB

S

2 1

p p p

Sv S v Sv

z p v p vSv S z p Sv

B

B B

B B B B

B B

2 1

2 1

1

2 1 1

= =

= +

= == +/ /

( / )p

v S Sz

p

vB B

2

2

1

1

1 0

1

=

/ ( )

TL LogS

S A cC

B

c

S

S D Log

cS

Szo o

o B

B

=

+ + +

= +10

1

4 10 1

2102

1

1

2

2

10

2


39/40





Helmholtz Resonator

z jS

M K

B

B

=

1

K c S

V

M S L

o B

o B

=

=

2 2

'

V

SB

L

L is the equivalentlength. Losses due toviscosity in the neck andradiation are neglected.

TL Log cSSz

o B

B

= + =10 1210

2

10 1 210 22

log/

'/ /+

c S

L S c VB


40/40


f K k 40


ME 599/699 Vibro-D

Summary 3

The transfer matrix method is based on plane wave (1-D)acoustic behavior (at component junctions).

The transfer matrix method can be used to determine thesystem behavior from component transfer matrices.

Applicability is limited to cascaded (series) componentsand simple branch components (not applicable to successivebranching and parallel systems).

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17_Design of Mufflers and Silencers2

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